Why does gibbs sampler work

Your jaw drops: the entry for each attack is about as large a the manual for the previous spell, because it lists a relative probability for each possible accompanying buff. Similarly, the probability of a particular attack spell occurring depends on the probability of the buff occurring.

It would be justified to wonder if a proper probability distribution can even be defined given this information. Well, it turns out that if there is one, it is uniquely specified by the conditional probabilities given in the manual. But how to sample from it? Luckily for you, the CD comes with an automated Gibbs' sampler, because you would have to spend an eternity doing the following by hand.

Choose a new attack spell using the accept-reject algorithm conditional on the buff in step 2. You see, in general, MCMC samplers are only asymptotically guaranteed to generate samples from a distribution with the specified conditional probabilities. But in many cases, MCMC samplers are the only practical solution available.

Gibbs sampling is not a self-contained concept. It requires some prerequisite knowledge. Below is the knowledge chain i summarized from my own study, as for your reference My major was applied physics :.

The document I named here is roughly following the chain. If the link is broken, google the document name. You will find it.

Some thoughts: I don't think Gibbs sampling can be understood solely by some abstracts. There is no shortcut for it. You need to understand the math behind it. And since it's more like a "technique", my criterion of "understanding it" is "you can edit its code and understand what you're doing not necessarily from scratch ".

For those who think they have understood it by looking at some quick notes, they probably just understand what is "Markov Chain Monte Carlo" in a high level and think they have got it all I made this illusion myself.

You can either run through them in a sequence, or you can randomly chose which of these to sample form. But you keep doing scans over and over to get a lot of samples. Why would you want this? That way we could use the law of large numbers and central limit theorems to approximate expectations, and we would have some idea of the error. And are they even identical are they even coming from the same distribution? Gibbs sampling can still give you a law of large numbers and a central limit theorem.

That means the marginal distribution of each draw is from the distribution you're targetting so they're identical draws. However, they are not independent.

In practice this means you run the chain for longer or you subsample the chain only take every th sample, say. Everything can still "work," though. For more information I would click the link underneath the question. There are some good references posted in that thread. This answer just attempts to give you the jist using the notation in common LDA references. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group.

Create a free Team What is Teams? Learn more. Can someone explain Gibbs sampling in very simple words? Asked 10 years, 6 months ago. Active 2 years, 4 months ago. Updating more variables at a time in blocks is helpful. Consider the limiting case where if we could sample from a block containing all of the variables, then we could sample directly from. Another variant is collapsed Gibbs sampling.

In this algorithm, we marginalize out as many variables as possible before sampling from the conditional distribution of some variable. In collapsed Gibbs sampling, we would alternately sample and then. Note that in this case, we are drawing samples from the exact distribution. Note that the ordering of the variables in the sampling procedure is very important for collapsed Gibbs sampling to ensure that the resulting Markov chain has the right stationary distribution since the right ordering might depend on which variables we marginalize out.

Simulated annealing is an adaptation of the Metropolis-Hastings algorithm and is a heuristic for finding the global maximum of a given function. Simulated annealing moves around the space trying to find assignments that satisfy as many clauses as possible.

We construct a probability distribution that puts high probability on assignments that satisfy many clauses. As , approaches the uniform distribution. As , tends to put all of the probability mass on satisfying assignments. To solve the optimization problem, we want to sample from this distribution when is small. We can use Metropolis-Hastings with a proposal distribution that randomly picks a variable and flips it.

Let be equal to if differ in only one variable and otherwise. The Metropolis-Hastings acceptance probability is. If , then we always accept the transition.